Catching a bus describes the process of getting to a stop/station, waiting and boarding. Taking a bus describes the entire process, including the journey itself. Much of the time, the distinction isn't important.

Yes, the phrase "catch a bus" is perfectly correct and widely used in both everyday conversation and formal writing. While "take the bus" is a more general and neutral term for using public transport, "catch a bus" often carries a subtle nuance of timing or urgency. It implies the act of arriving at the stop just in time to meet the vehicle's scheduled departure (e.g., "I need to run if I'm going to catch the 9:15 bus"). Linguistically, it is a "collocation" where the verb "catch" means to intercept or reach something that is in motion or on a schedule. You will frequently see it used in reputable news outlets like The New York Times or The Guardian. However, you should avoid using it in contexts where "boarding" or "riding" is more precise; for example, if you are already on the vehicle, you wouldn't say you are "catching" it. For very formal technical documents or transit announcements, you might see the term "board the bus" instead, but for almost any other situation, "catch a bus" is a natural and idiomatic choice in English.